15x^2 - 30x + 10 = 0
["solving the quadratic equation 15x² - 30x + 10 = 0: A Step-by-Step Guide for Students and Math Enthusiasts", "Understanding how to solve quadratic equations is essential in algebra and forms the foundation for many advanced math topics. Today, we’ll walk through the detailed process of solving the equation:", "15x² - 30x + 10 = 0", "Whether you're a student preparing for exams, a self-learner, or just curious about quadratics, this step-by-step explanation will help you solve this equation efficiently and understand the concepts behind it.", "---", "### What is a Quadratic Equation?", "A quadratic equation is a second-degree polynomial equation in the variable x, typically written in the standard form:", "$$\nax^2 + bx + c = 0\n$$", "where (a), (b), and (c) are constants, and (a <br/>\neq 0). The solutions (or roots) of the equation are found using the quadratic formula:", "$$\nx = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\n$$", "---", "### Step 1: Identify Coefficients", "Given the equation:\n15x² - 30x + 10 = 0", "We identify the coefficients:\n- (a = 15)\n- (b = -30)\n- (c = 10)", "---", "### Step 2: Compute the Discriminant", "The discriminant (D) determines the nature of the roots and is calculated as:", "$$\nD = b^2 - 4ac\n$$", "Plugging in the values:", "$$\nD = (-30)^2 - 4(15)(10) = 900 - 600 = 300\n$$", "Since (D = 300 > 0), there are two distinct real roots.", "---", "### Step 3: Apply the Quadratic Formula", "Using the quadratic formula:", "$$\nx = \frac{-(-30) \pm \sqrt{300}}{2 \cdot 15} = \frac{30 \pm \sqrt{300}}{30}\n$$", "---", "### Step 4: Simplify the Square Root", "$$\n\sqrt{300} = \sqrt{100 \cdot 3} = 10\sqrt{3}\n$$", "So the equation becomes:", "$$\nx = \frac{30 \pm 10\sqrt{3}}{30}\n$$", "---", "### Step 5: Simplify the Expression", "Divide numerator and denominator by 10:", "$$\nx = \frac{3 \pm \sqrt{3}}{3}\n$$", "This gives the two exact solutions:", "$$\nx = \frac{3 + \sqrt{3}}{3} \quad \ ext{and} \quad x = \frac{3 - \sqrt{3}}{3}\n$$", "Alternatively, simplifying each term:", "$$\nx = 1 + \frac{\sqrt{3}}{3} \quad \ ext{and} \quad x = 1 - \frac{\sqrt{3}}{3}\n$$", "Both forms are acceptable, but the first version is often preferred for clarity.", "---", "### Final Answer", "The solutions to the equation (15x^2 - 30x + 10 = 0) are:", "$$\n\boxed{x = 1 + \frac{\sqrt{3}}{3} \quad \ ext{and} \quad x = 1 - \frac{\sqrt{3}}{3}}\n$$", "---", "### Tips for Solving Quadratics Quickly", "- Always check the discriminant to determine how many real solutions exist.\n- Simplify square roots when possible for cleaner expressions.\n- Write the final answer in rationalized form if preferred.\n- Practice factoring when possible, but the quadratic formula guarantees solutions even when factoring is difficult.", "---", "### Why Learning This Matters", "Mastering quadratic equations enables you to model many real-world scenarios—from physics problems involving motion to business profit calculations—and strengthens your problem-solving skills. By understanding how to solve 15x² - 30x + 10 = 0, you gain confidence tackling more complex math problems ahead.", "---", "Key Takeaways:\n- Use the quadratic formula when factoring is challenging.\n- Simplify radicals to get clean, simplified answers.\n- Recognizing the nature of roots via the discriminant improves problem-solving insight.", "If you're looking to deepen your algebra skills or find reliable resources on solving quadratics, check out reputable math websites or online tutors offering step-by-step support.", "---", "Happy solving!\nwhether you’re nailing homework, preparing for exams, or just expanding your knowledge, remembering how to solve 15x² - 30x + 10 = 0 is a valuable step forward."]
